It reminds me of a cure for hiccups that I was told about as a child: Run three times around the house without thinking of wolves. It is very important not to think of wolves. If you do, the hiccups will continue. Anyway, to get more mathematical, just use the first positive zero of the sine function. No π needed.
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Harald Hanche-OlsenNov 13 '12 at 21:16

Why don't you inscribe the cylinder into a prism whose base is a regular polygon and has $h=z$, and then compute its volume as a function of $n$? The resulting expression does not involve $\pi$ and if you take the limit when $n$ goes to $\infty$ you get the volume of the cylinder. The tricky part is to express the apothema in terms of $n$, so you can check that the limit is actually finite, but it can be done.

To me, this sounds like a typical application of the density of $\mathbb{Q}$ in $\mathbb{R}$.

Sure you can without pi. Remove the top of the cylinder. Fill it with fine-grained sand.
Then pour the sand into a rectangular box, and measure how far it comes up. Of course, it will be an estimate, but not a bad one. (Get an even finer grade sand.) You can do the same by unscrewing the top of a sphere.